Advertisement

Algebra universalis

, Volume 73, Issue 3–4, pp 391–417 | Cite as

Commutative idempotent groupoids and the constraint satisfaction problem

  • Clifford Bergman
  • David Failing
Article

Abstract

A restatement of the Algebraic Dichotomy Conjecture, due to Maroti and McKenzie, postulates that if a finite algebra A possesses a weak near-unanimity term, then the corresponding constraint satisfaction problem is tractable. A binary operation is weak near-unanimity if and only if it is both commutative and idempotent. Thus if the dichotomy conjecture is true, any finite commutative, idempotent groupoid (CI groupoid) will be tractable. It is known that every semilattice (i.e., an associative CI groupoid) is tractable. A groupoid identity is of Bol–Moufang type if the same three variables appear on either side, one of the variables is repeated, the remaining two variables appear once, and the variables appear in the same order on either side (for example, \({x(x(yz)) \approx (x(xy))z}\)). These identities can be thought of as generalizations of associativity. We show that there are exactly 8 varieties of CI groupoids defined by a single additional identity of Bol–Moufang type, derive some of their important structural properties, and use that structure theory to show that 7 of the varieties are tractable. We also characterize the finite members of the variety of CI groupoids satisfying the self-distributive law \({x(yz) \approx (xy)(xz)}\), and show that they are tractable.

Key words and phrases

constraint satisfaction CSP dichotomy Bol–Moufang Plonka sum self-distributive squag quasigroup prover9 mace4 uacalc 

2010 Mathematics Subject Classification

Primary: 08A70 Secondary: 68Q25 08B25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barto, L., Kozik, M.: Constraint satisfaction problems of bounded width. In: 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), pp. 595–603. IEEE Computer Soc., Los Alamitos, CA (2009)Google Scholar
  2. 2.
    Bergman, C.: Universal Algebra: Fundamentals and Selected Topics. CRC Press, Boca Raton, FL (2012)Google Scholar
  3. 3.
    Bulatov, A., Dalmau, V.: A simple algorithm for Mal′tsev constraints. SIAM J. Comput. 36, 16–27 (electronic) (2006)Google Scholar
  4. 4.
    Bulatov, A., Jeavons, P.: Algebraic structures in combinatorial problems. Tech. rep., Technische Universitat Dresden (2001). MATH–AL–4–2001, http://www.cs.sfu.ca/~abulatov/papers/varieties.ps
  5. 5.
    Bulatov, A., Jeavons, P., Krokhin, A.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34, 720–742 (electronic) (2005)Google Scholar
  6. 6.
    Bulatov, A.A.: Combinatorial problems raised from 2-semilattices. J. Algebra 298, 321–339 (2006)Google Scholar
  7. 7.
    Bulatov, A.A., Valeriote, M.: Recent results on the algebraic approach to the CSP. In: N. Creignou, P.G. Kolaitis, H. Vollmer (eds.) Complexity of Constraints, pp. 68–92. Springer, Berlin (2008)Google Scholar
  8. 8.
    Burris, S., Sankappanavar, H.P.: A course in universal algebra, Graduate Texts in Mathematics, vol. 78. Springer, New York (1981)Google Scholar
  9. 9.
    Dalmau, V.: Generalized majority-minority operations are tractable. Log. Methods Comput. Sci. 2, 4:1, 14 pp. (2006)Google Scholar
  10. 10.
    Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: a study through Datalog and group theory. SIAM J. Comput. 28, 57–104 (electronic) (1999)Google Scholar
  11. 11.
    Fenyves, F.: Extra loops. II. On loops with identities of Bol-Moufang type. Publ. Math. Debrecen 16, 187–192 (1969)Google Scholar
  12. 12.
    Freese, R., Kiss, E., Valeriote, M.: Universal Algebra Calculator (2011). Available at: www.uacalc.org
  13. 13.
    Idziak, P., Marković, P., McKenzie, R., Valeriote, M., Willard, R.: Tractability and learnability arising from algebras with few subpowers. SIAM J. Comput. 39, 3023–3037 (2010)Google Scholar
  14. 14.
    Jeavons, P.: On the algebraic structure of combinatorial problems. Theoret. Comput. Sci. 200, 185–204 (1998)Google Scholar
  15. 15.
    Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. J. ACM 44, 527–548 (1997)Google Scholar
  16. 16.
    Ježek, J., Kepka, T.: The lattice of varieties of commutative abelian distributive groupoids. Algebra Universalis 5, 225–237 (1975)Google Scholar
  17. 17.
    Ježek, J., Kepka, T., Němec, P.: Distributive groupoids. Rozpravy Československé Akad. Věd Řada Mat. Přírod. Věd 91, 94 (1981)Google Scholar
  18. 18.
    Kepka, T.: Commutative distributive groupoids. Acta Univ. Carolin.—Math. Phys. 19(2), 45–58 (1978)Google Scholar
  19. 19.
    Kepka, T., Němec, P.: Commutative Moufang loops and distributive groupoids of small orders. Czechoslovak Math. J. 31(106), 633–669 (1981)Google Scholar
  20. 20.
    Kozik, M., Krokhin, A., Valeriote, M.A., Willard, R.: Characterizations of several Maltsev conditions. preprint (2013)Google Scholar
  21. 21.
    Kunen, K.: Quasigroups, loops, and associative laws. J. Algebra 185, 194–204 (1996). DOI  10.1006/jabr.1996.0321. URL http://dx.doi.org/10.1006/jabr.1996.0321
  22. 22.
    Mackworth, A.: Consistency in networks of relations. Artificial Intelligence 8(1), 99–118 (1977). Reprinted in Readings in Artificial Intelligence, B. L. Webber and N. J. Nilsson (eds.), Tioga Publ. Col., Palo Alto, CA, pp. 69–78, 1981.Google Scholar
  23. 23.
    McCune, W.: Prover9 and mace4 (2005–2010). http://www.cs.unm.edu/~mccune/prover9/
  24. 24.
    Mel′nik, I.I.: Normal closures of perfect varieties of universal algebras. In: Ordered sets and lattices, No. 1 (Russian), pp. 56–65. Izdat. Saratov. Univ., Saratov (1971)Google Scholar
  25. 25.
    Phillips, J.D., Vojtěchovský, P.: The varieties of loops of Bol-Moufang type. Algebra Universalis 54, 259–271 (2005)Google Scholar
  26. 26.
    Phillips, J.D., Vojtěchovský, P.: The varieties of quasigroups of Bol-Moufang type: an equational reasoning approach. J. Algebra 293, 17–33 (2005)Google Scholar
  27. 27.
    Płonka, J.: On a method of construction of abstract algebras. Fund. Math. 61, 183–189 (1967)Google Scholar
  28. 28.
    Płonka, J.: On equational classes of abstract algebras defined by regular equations. Fund. Math. 64, 241–247 (1969)Google Scholar
  29. 29.
    Quackenbush, R.W.: Varieties of Steiner loops and Steiner quasigroups. Canad. J. Math. 28, 1187–1198 (1976)Google Scholar
  30. 30.
    Romanowska, A.: On regular and regularised varieties. Algebra Universalis 23, 215–241 (1986)Google Scholar
  31. 31.
    Romanowska, A.B., Smith, J.D.H.: Modes. World Scientific Publishing Co. Inc., River Edge, NJ (2002)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Department of MathematicsQuincy UniversityQuincyUSA

Personalised recommendations